On the Sylow graph of a group and Sylow normalizers
Identifieur interne : 000277 ( Main/Exploration ); précédent : 000276; suivant : 000278On the Sylow graph of a group and Sylow normalizers
Auteurs : L. S. Kazarin [Russie] ; A. Martínez-Pastor [Espagne] ; M. D. Pérez-Ramos [Espagne]Source :
- Israel Journal of Mathematics [ 0021-2172 ] ; 2011-11-01.
Abstract
Abstract: Let G be a finite group and G p be a Sylow p-subgroup of G for a prime p in π(G), the set of all prime divisors of the order of G. The automiser A p (G) is defined to be the group N G (G p )/G p C G (G p ). We define the Sylow graph Γ A (G) of the group G, with set of vertices π(G), as follows: Two vertices p, q ∈ π(G) form an edge of Γ A (G) if either q ∈ π(A p (G)) or p ∈ π(A q (G)). The following result is obtained Theorem: Let G be a finite almost simple group. Then the graph Γ A (G) is connected and has diameter at most 5. We also show how this result can be applied to derive information on the structure of a group from the normalizers of its Sylow subgroups.
Url:
DOI: 10.1007/s11856-011-0138-x
Affiliations:
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S." last="Kazarin">L. S. Kazarin</name><affiliation wicri:level="1"><country xml:lang="fr">Russie</country><wicri:regionArea>Department of Mathematics, Yaroslavl P.Demidov State University, Sovetskaya Str 14, 150000, Yaroslavl</wicri:regionArea><wicri:noRegion>Yaroslavl</wicri:noRegion></affiliation><affiliation wicri:level="1"><country wicri:rule="url">Russie</country></affiliation></author><author><name sortKey="Martinez Pastor, A" sort="Martinez Pastor, A" uniqKey="Martinez Pastor A" first="A." last="Martínez-Pastor">A. Martínez-Pastor</name><affiliation wicri:level="1"><country xml:lang="fr">Espagne</country><wicri:regionArea>Escuela Técnica Superior de Ingeniería Informática Instituto Universitario de Matemática Pura y Aplicada IUMPA-UPV, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022, Valencia</wicri:regionArea><wicri:noRegion>Valencia</wicri:noRegion></affiliation><affiliation wicri:level="1"><country wicri:rule="url">Espagne</country></affiliation></author><author><name sortKey="Perez Ramos, M D" sort="Perez Ramos, M D" uniqKey="Perez Ramos M" first="M. D." last="Pérez-Ramos">M. D. Pérez-Ramos</name><affiliation wicri:level="1"><country xml:lang="fr">Espagne</country><wicri:regionArea>Departament d’` Algebra, Universitat de València, C/Doctor Moliner, 50, 46100, Burjassot (València)</wicri:regionArea><wicri:noRegion>Burjassot (València)</wicri:noRegion></affiliation><affiliation wicri:level="1"><country wicri:rule="url">Espagne</country></affiliation></author></analytic><monogr></monogr><series><title level="j">Israel Journal of Mathematics</title><title level="j" type="abbrev">Isr. J. Math.</title><idno type="ISSN">0021-2172</idno><idno type="eISSN">1565-8511</idno><imprint><publisher>The Hebrew University Magnes Press</publisher><pubPlace>Heidelberg</pubPlace><date type="published" when="2011-11-01">2011-11-01</date><biblScope unit="volume">186</biblScope><biblScope unit="issue">1</biblScope><biblScope unit="page" from="251">251</biblScope><biblScope unit="page" to="271">271</biblScope></imprint><idno type="ISSN">0021-2172</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0021-2172</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: Let G be a finite group and G p be a Sylow p-subgroup of G for a prime p in π(G), the set of all prime divisors of the order of G. The automiser A p (G) is defined to be the group N G (G p )/G p C G (G p ). We define the Sylow graph Γ A (G) of the group G, with set of vertices π(G), as follows: Two vertices p, q ∈ π(G) form an edge of Γ A (G) if either q ∈ π(A p (G)) or p ∈ π(A q (G)). The following result is obtained Theorem: Let G be a finite almost simple group. Then the graph Γ A (G) is connected and has diameter at most 5. We also show how this result can be applied to derive information on the structure of a group from the normalizers of its Sylow subgroups.</div></front></TEI><affiliations><list><country><li>Espagne</li><li>Russie</li></country></list><tree><country name="Russie"><noRegion><name sortKey="Kazarin, L S" sort="Kazarin, L S" uniqKey="Kazarin L" first="L. S." last="Kazarin">L. S. Kazarin</name></noRegion><name sortKey="Kazarin, L S" sort="Kazarin, L S" uniqKey="Kazarin L" first="L. S." last="Kazarin">L. S. Kazarin</name></country><country name="Espagne"><noRegion><name sortKey="Martinez Pastor, A" sort="Martinez Pastor, A" uniqKey="Martinez Pastor A" first="A." last="Martínez-Pastor">A. Martínez-Pastor</name></noRegion><name sortKey="Martinez Pastor, A" sort="Martinez Pastor, A" uniqKey="Martinez Pastor A" first="A." last="Martínez-Pastor">A. Martínez-Pastor</name><name sortKey="Perez Ramos, M D" sort="Perez Ramos, M D" uniqKey="Perez Ramos M" first="M. D." last="Pérez-Ramos">M. D. Pérez-Ramos</name><name sortKey="Perez Ramos, M D" sort="Perez Ramos, M D" uniqKey="Perez Ramos M" first="M. D." last="Pérez-Ramos">M. D. Pérez-Ramos</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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